Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

4.13 ATTENUATION COEFFICIENT

Back

The attenuation or dissipation of acoustic energy as a sound wave moves

through a medium may be attributed to three basic mechanisms:

(a) Viscous effects (dissipation of acoustic energy due to fluid friction),

which result in thermodynamically irreversible propagation

of sound.

(b) Heat conduction effects (heat transfer between high- and lowtemperature

regions in the wave), which result in non-adiabatic

propagation of the sound.

effects), which result in a time lag between changes in translational

kinetic energy and the energy associated with rotation

and vibration of the molecules.

The viscous energy dissipation effects result from the relative motion

between different portions of the fluid during compression and expansion

that occurs when a sound wave moves through the fluid. For a newtonian

fluid, the magnitude of this effect is proportional to the viscosity of the

fluid.

As the fluid is compressed and expanded during the transmission of a

sound wave, changes in temperature occur in different portions of the fluid.

Transmission of Sound 143

There is a tendency for energy to be conducted fromregions of compression,

where the temperature is elevated, to regions of expansion or rarefaction,

where the temperature is reduced. The heat transfer effect tends to reduce

the amplitude of the pressure wave and dissipate energy as the wave moves

through the medium. The magnitude of this effect is proportional to the

thermal conductivity kt of the fluid and inversely proportional to the specific

heat cp or the thermal energy storage capacity of the medium.

For all fluids except monatomic gases, there is a finite time lag for

conversion of energy into rotational and vibrational energy of the molecule.

During this time, the acoustic wave may move past the molecule and leave

behind some of the acoustic energy.

The attenuation due to the sum of the first two mechanisms, viscous

and heat conduction, is called the classical attenuation. The classical

attenuation may be written in the following form (Kinsler et al., 1982):

_рclassicalЮ ј

2_2 f 2

_c3

3 ю

_ _1

_ _

(4-221)

where ј fluid viscosity, _ ј specific heat ratio (1.667 for monatomic gases

and 1.400 for diatomic gases), Pr ј cp=kt ј Prandtl number, cp ј specific

heat at constant pressure, and kt ј thermal conductivity. The classical

relaxation time may be found from Eqs (4-211) and (4-221):

_рclassicalЮ ј

_c2

3 ю

_ _1

_ _

(4-222)

The classical attenuation coefficient is proportional to the square of the

frequency, and the classical relaxation time is independent of the frequency.

The expression for the classical attenuation coefficient yields good

agreement with experimental values of the attenuation coefficient for monatomic

gases, such as argon and helium, as shown in Table 4-7.

The third contribution to attenuation of sound in a fluid results from

the finite time requried to convert translational kinetic energy into internal

energies associated with rotation and vibration of the molecules. For many

molecules—including CO2, water vapor, nitrogen, and oxygen—vibrational

energy transfer is predominant in the acoustic frequency range. Theoretical

models, with experimental verification, have shown that the attenuation

coefficient can be written in terms of the sum of the individual contributions

(Pierce, 1988):

_ ј _рclassicalЮ ю __v;j (4-223)

where the _v;j are the contributions of the various vibrational energy relaxation

effects, which may be expressed in the form:

144 Chapter 4

_v ј

_1р!_vЮ2

1 ю р!_vЮ2 (4-224)

The quantity ! ј 2_f is the circular frequency of the sound wave, _v is the

relaxation time for vibrational energy, and _1 is the limiting (highfrequency)

value for the vibrational attenuation contribution.

The term _1 is related to the specific heat ratio _ for a gas and the

vibrational contribution to the specific heat, cvib:

_1 ј рcvib=RЮр_ _ 1Ю2

2c_v_

(4-225)

The vibrational specific heat term may be calculated from the following

expression (ter Haar, 1954):

cvib=R ј yjр_v=TЮ2 expр__v=TЮ (4-226)

The term yj is the mole fraction of the jth component in a gas mixture р yj ј 1 for a single component gas), _v is a constant, dependent on the gas, and T

is the absolute temperature of the gas. For nitrogen and oxygen, the constant

_v is as follows (ANSI, 1978):

_vрN2Ю ј 3352K

_vрO2Ю ј 2239K

The relaxation times for atmospheric air are sensitive to the amount of

water vapor present in the air. An O2 or N2 molecule colliding with an H2O

molecule is more likely to exhibit a change in vibrational energy than when

the molecules collide with other N2 or O2 molecules. The following expressions

may be used to estimate the relaxation times for oxygen and nitrogen

in atmospheric air (Sutherland et al., 1974):

Transmission of Sound 145

TABLE 4-7 Values of the Classical Attenuation Coefficient.

Gas

Viscosity,

_, mPa-s

Prandtl No.,

Pr ј _cp=kt

Relaxation

time, _, ns

_=f 2(calc.),

Np/m-Hz2

_=f 2(expt.),

Np/m-Hz2

Argon 22.73 0.668 0.314 19:20 _ 10

_12 19:40 _ 10

_12

Helium 19.94 0.692 0.271 5:25 _ 10

_12 5:35 _ 10

_12

Nitrogen 17.82 0.715 0.238 13:29 _ 10

_12 16:14 _ 10

_12

Oxygen 20.65 0.714 0.276 16:47 _ 10

_12 19:64 _ 10

_12

р pref=pЮ

2__vрO2Ю ј 24 ю р4:41Юр106Юh

0:05 ю 100h

0:391 ю 100h

(4-227)

р pref=pЮ

2__vрN2Ю ј Ѕ9 ю р3:5Юр104Юh e_F _рTref=TЮ1=2 (4-228)

F ј 6:142ЅрTref=TЮ1=3 _ 1_ (4-229)

The reference pressure and temperature values are pref ј 101:325 kPa and

Tref ј 293:16 K, respectively. The range of validity of Eqs (4-227) and

(4-228) is between 08C and 408C (328F and 1048F), for an accuracy within

10%. The quantity h is the fraction of molecules in the gas that are H2O

molecules. This fraction is related to the relative humidity RH, expressed as a

decimal (0.40 instead of 40%), and the saturation pressure of the water vapor

at the air temperature, psat:

h ј рRHЮ psat=p (4-230)

The energy attenuation coefficient for atmospheric air at various temperatures

and relative humidity values is presented in Table 4-8. The energy

attenuation coefficient m is related to the attenuation coefficient _ by

m ј 2_. The values given in Table 4-8 are values of the energy attenuation

coefficient averaged over the octave band with the indicated center frequency.

The octave band values are about 10% different from the values

146 Chapter 4

TABLE 4-8 Energy Attenuation Coefficient m (km_1) for

Atmospheric Air at 101.325 kPa (14.7 psia): Note that m ј 2_,

where _ is the Attenuation Coefficient.

Relative humidity,

Temperature,

Octave band center frequency, Hz

500 1,000 2,000 4,000 8,000

10 10 1.28 4.30 10.6 16.3 16.4

15 0.98 3.41 10.9 22.3 24.0

20 0.78 2.67 9.02 25.7 34.1

25 0.71 2.14 7.18 24.2 36.3

30 0.69 1.80 5.84 20.4 38.9

20 10 0.63 2.04 6.98 21.1 29.9

15 0.56 1.61 5.50 18.7 33.2

20 0.53 1.40 4.31 14.7 29.4

25 0.52 1.33 3.58 11.7 24.3

30 0.52 1.30 3.23 9.80 20.7

Transmission of Sound 147

Relative humidity,

Temperature,

Octave band center frequency, Hz

500 1,000 2,000 4,000 8,000

30 10 0.50 1.35 4.60 15.1 28.4

15 0.48 1.23 3.59 12.0 25.1

20 0.46 1.17 3.02 9.62 20.3

25 0.46 1.14 2.80 7.90 16.4

30 0.46 1.13 2.76 7.14 13.4

40 10 0.45 1.13 3.37 11.3 22.8

15 0.44 1.07 2.80 8.91 18.7

20 0.43 1.05 2.62 7.22 15.0

25 0.42 1.03 2.57 6.33 12.5

30 0.42 1.02 2.56 6.20 11.1

50 10 0.41 1.01 2.77 8.93 17.8

15 0.40 0.99 2.50 7.16 14.5

20 0.39 0.96 2.34 6.17 11.9

25 0.39 0.95 2.30 5.88 10.4

30 0.38 0.94 2.26 5.76 9.88

60 10 0.38 0.94 2.51 7.92 15.0

15 0.38 0.92 2.31 6.12 12.2

20 0.37 0.90 2.20 5.66 10.3

25 0.37 0.89 2.16 5.50 9.27

30 0.37 0.88 2.14 5.43 9.01

70 10 0.36 0.89 2.30 6.45 13.4

15 0.36 0.86 2.16 5.58 11.0

20 0.35 0.85 2.08 5.33 9.57

25 0.35 0.84 2.06 5.18 8.85

30 0.35 0.84 2.05 5.14 8.71

80 10 0.35 0.84 2.14 5.80 11.6

15 0.34 0.82 2.02 5.32 9.86

20 0.34 0.81 1.97 5.04 9.05

25 0.33 0.80 1.95 4.93 8.52

30 0.33 0.80 1.95 4.88 8.47

90 10 0.33 0.80 1.97 5.37 10.1

15 0.33 0.79 1.92 5.09 8.93

20 0.32 0.78 1.87 4.87 8.56

25 0.32 0.77 1.87 4.72 8.36

30 0.32 0.77 1.86 4.68 8.34

TABLE 4-8 (Cont’d)

evaluated at the center frequency, because the attenuation coefficient is not a

linear function of frequency.

Example 4-12. Determine the classical attenuation coefficient for argon

gas at 273.2K (08C or 328F) and 101.3 kPa (14.7 psia) at a frequency of

1000 Hz. The properties of argon gas at this condition are as follows:

viscosity ј 21:03 mPa-s (0.0509 lbm=ft-hr)

Prandtl number Pr ј 0:668

specific heat ratio _ ј 1:667

sonic velocity c ј 307:8 m/s (1010 ft/sec)

density _ ј 1:782 kg=m3 (0.1113 lbm=ft3Ю

The relaxation time may be found from Eq. (4-222):

_рclassicalЮ ј р21:03Юр10_6Ю

р1:782Юр307:8Ю2

3 ю р1:667 _ 1Ю

р0:668Ю

_ _

_рclassicalЮ ј р1:246Юр10_10Юр2:332Ю ј 0:290 _ 10_9 s ј 0:290 ns

Note that !_ ј р2_Юр1000Юр0:290Юр10_9Ю ј 1:825 _ 10_6 _ 1.

The attenuation coefficient may be calculated from Eq. (4-211),

because the quantity !_ is so small:

_рclassicalЮ ј р2_2Юр1000Ю2р0:290Юр10_9Ю

р307:8Ю ј 1:863 _ 10_5 Np=m

The attenuation coefficient may be expressed in decibel units.

8:6859_ ј р8:6859Юр1:863Юр10_5Ю ј 0:162 _ 10_3 dB=m ј 0:162 dB=km

Example 4-13. Calculate the energy attenuation coefficient for atmospheric

air at 293.2K (208C or 688F) and 101.3 kPa (14.7 psia) for a frequency

of 4 kHz. The relative humidity of the air is 20%. The properties of

air are as follows:

viscosity ј 18:21 mPa-s (0.0441 lbm/ft-hr)

Prandtl number Pr ј 0:717

specific heat ratio _ ј 1:400

sonic velocity c ј 343:2 m/s (1126 ft/sec)

density _ ј 1:204 kg=m3 (0.0752 lbm=ft3Ю

The saturation pressure for water vapor at 208C is psat ј 2:338 kPa.

First, let us calculate the contribution of viscous and thermal conduction

effects (the classical attenuation coefficient). The relaxation time for

viscous and thermal conduction effects is given by Eq. (4-222):

148 Chapter 4

_рclassicalЮ ј р18:21Юр10_6Ю

р1:204Юр343:2Ю2

3 ю р1:40 _ 1Ю

р0:717Ю

_ _

_рclassicalЮ ј 0:243 _ 10_9 s ј 0:243 ns

The classical attenuation coefficient may be calculated from Eq. (4-211):

_рclassicalЮ ј р2_2Юр4000Ю2р0:243Юр10_9Ю

р343:2Ю ј р13:97Юр10_9Юр4000Ю2

_рclassicalЮ ј 0:223 _ 10_3 Np=m

Next, let us calculate the effect of molecular interactions between the

O2 and H2O molecules. The fraction of air molecules at 20% relative humidity

that are water molecules is found from Eq. (4-230):

h ј р0:20Юр2:338Ю=р101:3Ю ј 0:00461

The relaxation time for the O2 interactions is found from Eq. (4-227):

р1Ю

р2_Ю_vрO2Ю ј 24 ю р4:41Юр106Юр0:00461Ю

0:05 ю 0:461

0:391 ю 0:461 ј 12,235 s_1

_vрO2Ю ј 1=р2_Юр12,235Ю ј 13:01 _ 10_6 s ј 13:01 ms

In atmospheric air, the mole fraction of oxygen is about yрO2Ю ј 0:21. The

vibrational specific heat terms may be determined from Eq. (4-226):

cvib=R ј р0:21Юр2239=293:2Ю2 expр_2239=293:2Ю ј 0:005904

The limiting attenuation coefficient for oxygen is found from Eq. (4-225),

using the sonic velocity for O2 of 326.6 m/s:

_1 ј р0:005904Юр1:40 _ 1Ю2

р2Юр326:6Юр13:01Юр10_6Юр1:40Ю ј 0:07941Np=m

For a frequency of 4000 Hz, we find the following:

!_vрO2Ю ј р2_Юр4000Юр13:01Юр10_6Ю ј 0:3269

The vibrational contribution associated with O2–H2O interactions may be

calculated from Eq. (4-224):

_vрO2Ю ј р0:07941Юр0:3269Ю2

1 ю р0:3269Ю2 ј 7:668 _ 10_3 Np=m

Let us repeat the calculations for the nitrogen and water vapor interactions.

The relaxation time is found from Eq. (4-228). The factor F ј 0

because T ј Tref in this example:

Transmission of Sound 149

р1Ю

2__vрN2Ю ј р1ЮЅ9 ю р3:5Юр104Юр0:00461Ю_ ј 170:5 s_1

_vрN2Ю ј 1=р2_Юр170:5Ю ј 0:933 _ 10_3 s ј 0:933 ms

The mole fraction of nitrogen in atmospheric air is yрN2Ю ј 0:79. The vibrational

specific heat term for N2–H2O interactions is as follows:

cvib=R ј р0:79Юр3352=293:16Ю2 expр_3352=293:16Ю ј 0:001118

The limiting attenuation coefficient for nitrogen—with a specific heat ratio

_ ј 1:40 and a sonic velocity of 349.0 m/s—is as follows:

_1 ј р0:001118Юр1:40 _ 1Ю2

р2Юр349:0Юр0:933Юр10_3Юр1:40Ю ј 0:0001961Np=m

At a frequency of 4000 Hz, we find the following for N2:

!_vрN2Ю ј р2_Юр4000Юр0:933Юр10_3Ю ј 23:45

The vibrational contribution associated with N2–H2O interactions is as

follows:

_vрN2Ю ј р0:0001961Юр23:45Ю2

1 ю р23:45Ю2 ј 0:196 _ 10_3

The attenuation coefficient is composed of the components that we

have calculated:

_ ј _рclassicalЮ ю Ѕ_vрO2Ю ю _vрN2Ю_

_ ј Ѕ0:223 ю р7:668 ю 0:196Ю_р10_3Ю ј 8:087 _ 10_3 Np=m

2:8%ю 94:8%ю 2:4%

The most important contribution to the attenuation coefficient is the internal

vibrational energy interactions for the oxygen molecules (almost 95% of

the total).

The attenuation coefficient may be expressed in decibel ‘‘units’’:

8:6859_ ј 0:0702 dB=m

The energy attenuation coefficient is as follows:

m ј 2_ ј р2Юр0:008087Ю ј 0:01617m_1

4:3429m ј 0:0702 dB=m

It is noted from Eqs (4-221) and (4-222) that the viscous and thermal

conduction effects result in a classical attenuation that is proportional to the

frequency squared, or the attenuation coefficient increases at a rate of 6 dB/

octave:

150 Chapter 4

_рclassicalЮ=f2 ј 2_2_рclassicalЮ=c ј constant

For the case, р2_f _vЮ > 10, the vibration contribution to the attenuation is

approximately constant with frequency (with 1%), as shown by Eq. (4-224):

_v!_1 ј constant (for !_v > 10Ю

On the other hand, for the case, р2_f _vЮ < 10, the vibrational contribution

is approximately proportional to the frequency squared.

_v=f2!4_2_1_2

v ј constant (for !_v < 10Ю

Example 4-14. A gas turbine has a sound power output spectrum as given

in Table 4-9. The directivity factor may be taken as Q ј 2 for all frequencies.

The noise is transmitted through atmospheric air at 258C (778C or

298.2K) and 101.3 kPa (14.7 psia) with a relative humidity of 50%. A residence

is located 400m (1312 ft) from the gas turbine unit. Determine the

overall sound pressure level at the residence location due to the turbine

noise.

The acoustic intensity for each octave band is given by Eq. (4-218),

with the directivity factor included:

I ј

QWo e_2_r

4_r2 ј

QWo e_mr

4_r2

The results of the calculations are summarized in Table 4-9. Let us present

the calculations for the 2000 Hz octave band. The acoustic power at the

source (gas turbine) is as follows:

Wo ј Wref 10Lw=10 ј р10_12Юр1012:4Ю ј 2:512W

The factor involving the energy attenuation coefficient for the 2000 Hz

octave band—note that the energy attenuation coefficient is given in units

of km_1—is as follows:

expЅ_mr_ ј expЅ_р2:30Юр0:400Ю_ ј 0:3985

Attenuation of sound by atmospheric air reduces the acoustic power by a

factor of almost 0.40. The acoustic intensity at the receiver position (at the

residence) for the 2000 Hz octave band is as follows:

I ј р2Юр2:512Юр0:3985Ю

р4_Юр400Ю2 ј 0:996 _ 10_6W=m2 ј 0:996 mW=m2

The calculations may be repeated for the other octave bands. The

overall intensity is the sum of the intensities in each octave band:

Transmission of Sound 151

152 Chapter 4

TABLE 4-9 Solution for Example 4-14.

Octave band center frequency, Hz

63 125 250 500 1,000 2,000 4,000 8,000

Turbine LW, dB 120 124 128 128 127 124 123 123

Wo, watts 1.000 2.512 6.310 6.310 5.012 2.512 1.995 1.995

m, km

_1 0.0093 0.037 0.15 0.39 0.95 2.30 5.88 10.4

_mr 0.9963 0.9853 0.9418 0.8556 0.6839 0.3985 0.0952 0.0156

I, mW/m2 0.991 2.462 5.911 5.370 3.409 0.996 0.189 0.031

Io ј _I ј р0:991ю2:462ю5:911ю_ _ _Юр10_6Ю

Io ј 19:359_10_6W=m2 ј 19:359 mW=m2

The overall sound pressure level is calculated from the intensity as follows:

p ј р_ocIoЮ1=2 ј Ѕр409:8Юр19:359Юр10_6Ю_1=2 ј 0:0891 Pa

The overall sound pressure level is found as follows:

Lp ј 20 log10р0:0891=20_10_6Ю ј 73:0dB

We will show how these calculations can be carried out directly in terms of

decibels in Chapter 5.

It may be noted fromTable 4-9 that the effect of atmospheric attenuation

is practically negligible (e_mr > 0:90 or 1 _ e_mr < 0:10) in the 63 Hz,

125 Hz, and 250 Hz octave bands. On the other hand, the attenuation is

significant (e_mr < 0:10Ю in the 4 kHz and 8 kHz octave bands. From this

result, we may conclude that we are generally justified in neglecting atmospheric

air attenuation at low frequencies (below about 500 Hz), unless the

distance from the source is large. For a distance of 400m (1312 ft or about

4 mile), the reduction in the intensity due to atmospheric attenuation is

e_mr ј 0:0156 _ 1=64. On the other hand, for a distance of 4m (13.1 ft),

the factor e_mr ј 0:959 for the 8 kHz octave band. We can conclude that

the effect of attenuation in atmospheric air is also negligible when the sound

is transmitted over relatively small distances. If we set the ‘‘negligible’’ limit

at less than 0.5 dB, then atmospheric attenuation may be neglected when the

following condition is valid:

4:3429mr _ 0:5dB

mr _ 0:12

Пресс-релиз популярных книг

4.13 ATTENUATION COEFFICIENT

Популярные книги

Популярные статьи

Навигация